Habits of Grains in Dense Polycrystalline Solids

David M. Saylora, Bassem El Dasher, Ying Pang, Herbert M. Miller, Paul Wynblatt,

Anthony D. Rollett, and Gregory S. Rohrer*

Department of Materials Science and Engineering, Carnegie Mellon University,

Pittsburgh, Pennsylvania 15213-3890aNational Institute of Standards and Technology, Gaithersburg, MD 20899, USA

AbstractWe show here that the boundaries of individual grains in dense polycrystals prefer

certain crystallographic habit planes, almost as if they were independent of the

neighboring crystals. In MgO, SrTiO3, MgAl2O4, TiO2, and Al, the specific habit planes

within the polycrystal correspond to the same planes that dominate the external growth

forms and equilibrium shapes of isolated crystals of the same phase. The observations

reduce the apparent complexity of interfacial networks and suggest that the mechanisms

of solid state grain growth may be analogous to conventional crystal growth. The results

also indicate that a model for grain boundary energy and structure based on grain surface

relationships is more appropriate than the widely accepted models based on lattice

orientation relationships.

*contact author and member, American Ceramic Society.This work was supported by the MRSEC program of the National Science Foundationunder Award Number DMR-0079996.Version 11.07.03: Journal of the American Ceramic Society, in press.

Saylor et al. Habits of Grains …. p. 2

I. Introduction

The physical properties of polycrystalline solids depend on the structure of the

three-dimension network of internal interfaces. With a few notable exceptions, most of

what we know about the network structure and the distribution of three-dimensional grain

shapes has been derived from foams or computer simulations with isotropic boundary

properties1-3. The purpose of this communication is to report recent measurements of the

crystallographic distribution of internal grain surface orientations.

Five parameters are needed to characterize grain boundaries in polycrystalline

solids: three can be associated with the lattice misorientation and two with the orientation

of the interface plane. Using electron backscattered diffraction (EBSD) mapping, it is

possible to measure four of the five parameters from a single section plane. The fifth

parameter, the inclination of the boundary with respect to the section plane, can be

determined either by serial sectioning4,5 or stereological analysis6,7. Recent measurements

of the distribution of grain boundaries in polycrystalline MgO indicate that grains are

most frequently bounded by low energy {100} surface planes5,8. Furthermore, the

variation of the grain boundary energy with type is, to first order, simply proportional to

the variation of the sum of the energies of the two surfaces that comprise the boundary.

Provided that this approximation holds generally in polycrystalline solids, an extremely

useful simplifying principle emerges: knowledge of the surface energy anisotropy, which

depends on two parameters, is sufficient for predicting the anisotropy of the grain

boundary energy. This is useful because surface energies are easier to measure than grain

boundary energies. A more recent study of SrTiO3 (cubic, perovskite structure) also

showed that the lowest energy surfaces dominate the distribution of grain boundary

Saylor et al. Habits of Grains …. p. 3

planes9. In the current paper we add to the earlier findings new results from TiO2

(tetragonal, rutile structure), MgAl2O4 (cubic, spinel structure), and Al (cubic close

packed structure). The picture that emerges is that the phenomenon originally observed

in MgO can be generalized to other polycrystals: the most frequently observed grain

boundary planes are the same planes that dominate the external growth forms and

equilibrium shapes of isolated crystals of the same phase.

II. Experimental

Before analysis, each sample was annealed at a high homologous temperature to

promote grain growth. Aspects of the sample origin, preparation, and characteristics are

summarized in Table I. Crystal orientation maps on planar sections were obtained using

an EBSD mapping system (TexSEM Laboratories, Inc.) integrated with a scanning

electron microscope (Phillips XL40 FEG)12. The orientation maps contained between

5000 and 30000 grains and had a spatial resolution of ≤ 2 mm. The internal grain surface

orientations of SrTiO3 were determined by analyzing orientation data from two parallel

sections, as described previously9. For the other specimens, the distributions of internal

grain surfaces were determined by a stereological analysis of the lines of intersection

between the grain boundary plane and specimen surface6,7. The lines of intersection were

determined directly from the orientation maps using a procedure described by Wright and

Larsen13. In the current communication, we describe only the distribution of internal

grain surfaces, indexed in the crystal reference frame, averaged over all misorientations.

The procedure for this analysis has been described previously5. The results are presented

in multiples of a random distribution (MRD); values greater than one indicate planes

Saylor et al. Habits of Grains …. p. 4

observed more frequently than expected in a random distribution. The resolution of the

distribution is approximately 10 °.

III. Results

The preferred orientations for grain boundary planes in each polycrystal are

illustrated in Fig. 1. In SrTiO3, internal {100} planes are found with twice the frequency

of {111} planes9. Calculations14,15 and experimental measurements10 agree that the (100)

plane has the minimum energy. For the case of MgAl2O4, the data show that internal

{111} planes occur with more than twice the frequency of {100} planes. This is in

agreement with the free surfaces of naturally occurring spinels, which are typically

dominated by {111} planes16. The most frequently observed twin and cleavage plane in

spinel is also (111). Surface energy calculations, on the other hand, have yielded

ambiguous results17,18. Early calculations concluded that the (111) plane has the lowest

energy17, while more recent calculations indicate a minimum at the (100) orientation18.

Even so, the habits of the grains in the polycrystal are consistent with the habits of spinel

crystals observed in natural settings.

Tetragonal TiO2 (rutile) crystals display {110}, {100}, {101}, and {111} facets.

The cleavage planes are {110} and {100} and twins are found on {101} planes19.

Calculations identify the minimum energy surface as (110), while the (100) and (101)

surfaces have higher energies20. The internal surfaces of the rutile polycrystal show the

same trend (see Fig. 1c): {110} planes are the most abundant, followed by {100}, {111},

and {101}. This is the only noncubic system investigated so far and the uniaxial

symmetry about the [001] direction is distinct from the other three distributions. It is

noteworthy that the minimum in the distribution occurs at the (001) orientation,

Saylor et al. Habits of Grains …. p. 5

consistent with its absence on growth habits and the theoretical result that it has the

highest energy20.

The cohesive forces in MgO, SrTiO3, MgAl2O4, and TiO2 are dominated by long

range electrostatic interactions. Aluminum was examined as an example of a crystalline

material with short range cohesive interactions. Here, internal {111} planes are

preferred. The (111) surface has the highest coordination number of any plane in this

structure and observations of microscopic cavities in aluminum suggest that this is the

plane of minimum energy21. The predominance of internal planes with low surface

energy, observed in all of the materials, suggests that the grain boundaries made up of

these planes also have a low energy. The grain boundary energy must be equal to the

sum of the energies of the two free surfaces on either side of the boundary, minus a

binding energy that arises from the bonds formed when the two surfaces are brought

together22. Thus, the current observations suggest that variations in the binding energy

are small compared to the variations in the surface energy. The relationship between

surface energy and grain boundary population is also reflected in the magnitude of the

deviation from random in the distributions of internal surfaces. Aluminum, which is

thought to have the smallest surface energy anisotropy of the materials studied, deviates

less from random than the other materials. These findings are consistent with previous

reports that the grain boundary population is correlated more strongly to the grain

boundary plane orientation than to the lattice misorientation8,9.

IV. Discussion

The average grain habits described here should be thought of as growth forms

rather than equilibrium shapes. Therefore, it would not be justified to assume an exact

Saylor et al. Habits of Grains …. p. 6

inverse correlation between the observed distribution of planes and the surface energy

anisotropy. However, we should also recognize that the slow growing faces found on

crystal habits typically correspond to low energy surfaces and, in this way, we can

explain the tendency for certain low energy planes to be preferred on grain shapes.

While the results show that grains in polycrystals display preferred habit planes,

the topology of the network demands the simultaneous presence of non-habit planes. For

example, since the average number of faces on a grain (13-14) is greater than the

multiplicity of the habit planes, and these planes have some curvature, it is necessary to

introduce non-habit planes in the interfacial network. Furthermore, if there is a low index

plane on one side of the boundary, then the plane on the other side is determined by the

lattice misorientation and for an arbitrary misorientation, the adjoining surface is most

likely to be a non-habit plane. Because of these geometric constraints, it would be

incorrect to think of the polycrystal as a collection of self-similar shapes. The preferred

grain habits and the topological requirements do, however, constrain the possible crystal

arrangements and this will affect all materials properties that depend on the interfacial

connectivity. Furthermore, any physical property that is anisotropic will also be

influenced by these constraints on local arrangements23.

The fact that internal grain surfaces have the tendency to take the same

orientations as external crystal surfaces suggests that grains in polycrystals might grow

by mechanisms analogous to those of crystals growing in a vapor or liquid phase. This

idea has a significant impact on the conventional theory of grain growth which assumes

that the rate of growth or shrinkage of a grain is determined only by its size24. In fact, the

rate at which a given grain grows or shrinks will be influenced by the fraction of its

Saylor et al. Habits of Grains …. p. 7

bounding area made up of slow moving habit planes. Elementary theories of grain

boundary motion depict thermally activated atom transfer across the interface25.

According to this picture, mobility is controlled by the availability of donor and receptor

sites on either side. Low energy facets tend to have low densities of such sites (with

possible additional barriers to the thermally activated formation of such sites) and may

therefore control the mobility, independent of the facet on the other side of the boundary.

V. Conclusions

Observations of internal surfaces in MgO, SrTiO3, MgAl2O4, TiO2, and Al lead us

to conclude that the habit of a phase in contact with itself (but misoriented) is not

substantially different from its habit when in contact with a gas or liquid phase.

Therefore, if the surface energy anisotropy of a phase, or even its natural habit, is known,

it is possible to predict the distribution of internal interfaces in a dense polycrystal of the

same material. The results also suggest a simple relationship between the surface and

grain boundary energy and indicate that solid state grain growth might occur by

mechanisms analogous to those which govern the growth of crystals in liquid or vapor

phases.

Saylor et al. Habits of Grains …. p. 8

References

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Phase Field Model,” Acta Mater. 50 [12] 3057-73 (2002).

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74 (2003).

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press.

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51 [13] 3675-86 (2003).

Saylor et al. Habits of Grains …. p. 9

9. D.M. Saylor, B.S. El-Dasher, T. Sano, and G.S. Rohrer, "Distribution of Grain

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Soc., submitted.

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1400 °C in Air," J. Amer. Ceram. Soc., 86 [11] (2003). in press.

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12. The naming of commercial equipment does not imply endorsement by NIST.

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Microscopy Scan Data,” J. Micro. 205 [3] 245-52 (2002).

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438.

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4576-80 (1977).

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Energy of Spinel MgAl2O4,” J. Am. Ceram. Soc. 83 [8] 2082-84 (2000).

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Saylor et al. Habits of Grains …. p. 10

20. M. Ramamoorthy, D. Vanderbilt, and R.D. King-Smith, “First-Principles

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16721-27 (1994).

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Saylor et al. Habits of Grains …. p. 11

Figure Caption

Figure 1. The distribution of grain boundary planes averaged over all misorientations for

(a) SrTiO3, (b) MgAl2O4, (c) TiO2, (d) Al. The data are shown in stereographic

projection along [001], which is in the center of each projection. The [110] direction is

marked with a square in each projection. In (c), the directions normal to the {101}

surfaces are marked with white diamonds and the directions normal to the {111} surfaces

are marked with white circles.

Table I. Materials

Material Space group grain size (mm) final heat, ° C/h Source/preparation

SrTiO3

†

Pm3 m 90 1650, 48 h See reference 10

MgAl2O4

†

Fd3 m 12 1600, 48 h RCS Technologies,sintered disk.

TiO2

†

P42 /mnm 55 1600, 24 h See reference 11

Al

†

Fm3 m 30 400, 1 h Alcoa, commerciallypure alloy 1050

Saylor et al. Habits of Grains …. p. 12

Figure 1. The distribution of grain boundary planes averaged over all misorientations for(a) SrTiO3, (b) MgAl2O4, (c) TiO2, (d) Al. The data are shown in stereographic

projection along [001], which is in the center of each projection. The [110] direction ismarked with a square in each projection. In (c), the directions normal to the {101}

surfaces are marked with white diamonds and the directions normal to the {111} surfaces

are marked with white circles.

MRD

(a) SrTiO3

MRD

(b) MgAl2O4

MRD

(d) Al

MRD

(c) TiO2